The distributions and diffusivities of small ions in chondroitin sulphate, hyaluronate and some proteoglycan solutions

The distributions and diffusivities of Na+, Ca2+ and Cl- in chondroitin sulphate (CS), hyaluronate (HA) and proteoglycan solutions were measured using equilibrium dialysis and a capillary tube method. Measurements were made for a range of glycosaminoglycan (GAG) concentrations up to those normally found in dense connective tissue (10% CS, 2.5% HA), ionic strengths up to normal physiological concentrations (0.15 M) and for different combinations of monovalent and divalent cations. The partition coefficients, Ki, of the positive ions increased with increasing matrix concentration and with decreasing ionic strength but with one exception the selectivity coefficient KCaNa = square root of KCa/KNa was close to unity, indicating nearly ideal Donnan distributions. The ionic diffusivities decreased very much like those of small neutral solutes with increasing matrix concentration and with one exception were relatively independent of ionic strength, The exception in both cases was low matrix concentrations and low ionic strengths for which the diffusivity of Ca2+ was an order of magnitude lower and selectivity coefficients were approximately 2. We conclude that at physiological ionic strengths and GAG concentrations the distributions of small ions are determined by simple electrostatic interactions, without binding or condensation, and the diffusivities are not affected by the electrostatic field.     

The theoretical distributions and diffusivities of small ions in chondroitin sulphate and hyaluronate

The electrostatic interactions between polyionic glycosaminoglycans and small mobile ions are investigated using the Poisson-Boltzmann equation and a rod-in-cell model of the polyelectrolyte. Calculations are made for the range of polyelectrolyte concentrations and buffer compositions for which measurements of ion distributions and diffusivities are reported in a companion paper (Maroudas et al., Biophys. Chem. 32 (1988) 257). We conclude that the distribution of mobile ions is largely determined by the 'far-field' potential and is adequately described by the Poisson-Boltzmann theory and also by more approximate theories such as ideal Donnan or 'condensation' theory. The measured variations in cation diffusivities, particularly the increase in diffusivity with increasing matrix concentration at low ionic strengths, are predicted qualitatively using an approximate diffusion theory together with the calculated potential fields. However, the same theory applied to anion diffusion gives qualitatively wrong results.     

Electric charge distribution in proteins and ionic strength dependence of reaction rate

The ionic strength dependence of the reaction rate between protein and dichloride anion radical has been investigated by flash photolysis of aqueous chloride-containing lysozyme, ribonuclease A, or insulin. The rate constant for the reaction of lysozyme or ribonuclease A with dichloride anion radicals decreases with increasing ionic strength, while it increases for insulin. The dependence was found to obey an equation derived from the theory of Debye and Hückel or the equation of Wherland and Gray for lysozyme within experimental errors. For ribonuclease A, however, it deviates largely from these equations. In the case of insulin a moderate deviation was observed. The different behavior in the ionic strength dependence is discussed in terms of the electric charge distribution in the protein molecules.     

pH influence on enzymic activity: the involvement of two active ionized forms of either substrate or enzyme in the reaction.

The interpretation of graphs of pKm as a function of pH by the theory of Dixon (Dixon, 1953) is based on the assumption that each component of the reaction system is active in a particular ionic state and completely inactive in other ionic states. In this paper we analyse the case in which two ionic forms of either the substrate or the enzyme are active to different degrees. After definition of reaction system of this type, a generalized equation is derived connecting the reaction rate to the total substrate concentration. The pH dependence of Km deduced from this equation is discussed and compared with that deriving from the theory of Dixon. The present model predicts that the pKm dependence on pH has the form of a “wave” with two inflexion points, thus providing an alternative interpretation of such behaviour to that afforded by Dixon's formulation.     

Effects of ionic strength on lysozyme uptake rates in cation exchangers. I: Uptake in SP Sepharose FF

Fluorescence scanning confocal microscopy was used in parallel with batch uptake and breakthrough measurements of transport rates to study the effect of ionic strength on the uptake of lysozyme into SP Sepharose FF. In all cases the adsorption isotherms were near-rectangular. As described previously, the intraparticle profiles changed from slow-moving self-sharpening fronts at low salt concentration, to fast-moving diffuse profiles at high salt concentration, and batch uptake rates correspondingly increased with increasing salt concentration. Shrinking core and homogeneous diffusion frameworks were used successfully to obtain effective diffusivities for the low salt and high salt conditions, respectively. The prediction of column breakthrough was generally good using these frameworks, except for low-salt uptake results. In those cases, the compressibility of the stationary phase coupled with the shrinking core behavior appears to reduce the mass transfer rates at particle-particle contacts, leading to shallower breakthrough curves. In contrast, the fast uptake rates at high ionic strength appear to reduce the importance of mass transfer limitations at the particle contacts, but the confocal results do show a flow rate dependence on the uptake profiles, suggesting that external mass transfer becomes more limiting at high ionic strength. These results show that the complexity of behavior observable at the microscopic scale is directly manifested at the column scale and provides a phenomenological basis to interpret and predict column breakthrough. In addition, the results provide heuristics for the optimization of chromatographic conditions.     

Immobilized enzymes: electrokinetic effects on reaction rates in a porous medium

The effect of the internal diffusion and electrical surface charge on the overall rate of a reaction catalyzed by an enzyme immobilized on a porous medium are examined. Effectiveness factors have been calculated which compare the global reaction rate to that existing in the absence of the internal diffusion and/or the electrical field. The surface charge, assumed to arise from the dissociation equilibria of the acidic and basic surface groups of the enzyme, generates an electrical double layer at the pore surface. The double-layer potential is governed by the Poisson-Boltzmann equation. It is shown that the diffusion potential can be characterized by a modulus which depends upon the surface reaction rate, the charges and diffusivities of the substrate and products, the ionic strength, and the pore dimensions. The flux of a charged species in the pore occurs under the influences of the concentration gradient and the electrical potential gradient. The governing equations are solved by an iterative numerical method. The effects of pH, enzyme concentration, and substrate concentration on the rates of two different hydrolysis reactions catalyzed by immobilized papain are examined. The release of H(+) in one of the reactions causes the lowering of internal pH, and also a constancy of the internal pH when the external pH in creases beyond a certain value. The latter reaction also shows a maximum in the reaction rate with respect to enzyme concentration. The reaction not involving H(+) as a product shows a maximum in the reaction rate with respect to external pH, but a monotonic increase in the reaction rate as the enzyme concentration increases.     

Kinetics of electron transfer from thioredoxin reductase to thioredoxin

The reduction of Escherichia coli thioredoxin by thioredoxin reductase was studied by stopped-flow spectrophotometry. The reaction showed no dependence on thioredoxin concentration, indicating that complex formation was rapid and occurred during the dead time of the instrument. The kobs for the reaction of approximately 20 s-1 probably reflects the rate of electron transfer from thioredoxin reductase to thioredoxin and agrees with the kcat observed by steady-state kinetics. The reaction rate was unaffected by increasing the ionic strength, suggesting a lack of electrostatic stabilization in the interaction of the two proteins. A mutant thioredoxin in which a positively charged lysine in the active-site region was changed to a glutamic acid residue resulted in an electrostatic destabilization. Thioredoxin K36E was still a substrate for the reductase, but binding was impaired so that the rate could be measured by stopped-flow techniques as reflected by a dependence on protein concentration. Raising the ionic strength in this reaction served to shield the negative charge and increased the rate of binding to the reductase.     

Immobilized enzymes: Electrokinetic effects on reaction rates under external diffusion

The rates of reactions catalyzed by enzymes immobilized on a nonporous solid surface have been computed employing a Nernst film model. The Nernst-Planck equations for the transport of the charged substrate and product species in the film and the Poisson equation for the distribution of electrical potential are solved numerically with the appropriate boundary conditions. The electrical charge at the surface is assumed to arise from the dissociation equilibria of the acidic and basic surface groups of the enzyme. The pH at the surface affects both the surface charge as well as the intrinsic kinetics of the enzyme-catalyzed reaction. Factors which determine the pH at the surface include the pH in the bulk solution and the release of H(+) ions in the enzyme-catalyzed reaction. The latter causes a lowering of pH at the surface, causing the reaction rate to differ from that computed assuming an equilibrium distribution of electrical potential. Another kind of nonequilibrium contribution is caused by unequal charges or diffusivities of the substrate and products, which results in a diffusion potential being set up. Two moduli are introduced to evaluate the significance of the reaction-generated lowering of pH and the diffusion potential effect. The effect of changing various parameters, e.g., reaction rate constant, substrate concentration, enzyme concentration, pH, etc., on the overall reaction rate are studied.     

A theoretical model successfully identifies features of hepatitis B virus capsid assembly.

The capsids of most spherical viruses are icosahedral, an arrangement of multiples of 60 subunits. Though it is a salient point in the life cycle of any virus, the physical chemistry of virus capsid assembly is poorly understood. We have developed general models of capsid assembly that describe the process in terms of a cascade of low order association reactions. The models predict sigmoidal assembly kinetics, where intermediates approach a low steady state concentration for the greater part of the reaction. Features of the overall reaction can be identified on the basis of the concentration dependence of assembly. In simulations, and on the basis of our understanding of the models, we find that nucleus size and the order of subsequent "elongation" reactions are reflected in the concentration dependence of the extent of the reaction and the rate of the fast phase, respectively. The reaction kinetics deduced for our models of virus assembly can be related to the assembly of any "spherical" polymer. Using light scattering and size exclusion chromatography, we observed polymerization of assembly domain dimers of hepatitis B virus (HBV) capsid protein. Empty capsids assemble at a rate that is a function of protein concentration and ionic strength. The kinetics of capsid formation were sigmoidal, where the rate of the fast phase had second-power concentration dependence. The extent of assembly had third-power concentration dependence. Simulations based on the models recapitulated the concentration dependences observed for HBV capsid assembly. These results strongly suggest that in vitro HBV assembly is nucleated by a trimer of dimers and proceeds by the addition of individual dimeric subunits. On the basis of this mechanism, we suggest that HBV capsid assembly could be an important target for antiviral therapeutics.     

Hybrid MD-Nernst Planck model of α-hemolysin conductance properties

Motivated by experiments in which an applied electric field translocates polynucleotides through an α-hemolysin protein channel causing ionic current transient blockade, a hybrid simulation model is proposed to predict the conductance properties of the open channel. Time scales corresponding to ion permeation processes are reached using the Poisson–Nernst–Planck (PNP) electro-diffusion model in which both solvent and local ion concentrations are represented as a continuum. The diffusion coefficients of the ions (K⁺ and Cl^?) input in the PNP model are, however, calculated from all-atom molecular dynamics (MD). In the MD simulations, a reduced representation of the channel is used. The channel is solvated in a 1?M KCl solution, and an external electric field is applied. The pore specific diffusion coefficients for both ionic species are reduced 5–7 times in comparison to bulk values. Significant statistical variations (17–45%) of the pore-ions diffusivities are observed. Within the statistics, the ionic diffusivities remain invariable for a range of external applied voltages between 30 and 240?mV. In the 2D-PNP calculations, the pore stem is approximated by a smooth cylinder of radius ～9?Å with two constriction blocks where the radius is reduced to ～6?Å. The electrostatic potential includes the contribution from the atomistic charges. The MD-PNP model shows that the atomic charges are responsible for the rectifying behaviour and for the slight anion selectivity of the α-hemolysin pore. Independent of the hierarchy between the anion and cation diffusivities, the anionic contribution to the total ionic current will dominate. The predictions of the MD-PNP model are in good agreement with experimental data and give confidence in the present approach of bridging time scales by combining a microscopic and macroscopic model.     

A spectroscopic and thermodynamic study of porphyrin/DNA supramolecular assemblies.

Assemblies of trans-bis(N-methylpyridinium-4-yl)diphenylporphine ions on the surface of calf thymus DNA have been studied using several spectroscopic techniques: absorbance, circular dichroism, and resonance light scattering. The aggregation equilibrium can be treated as a two-state system-monomer and assembly-each bound to the nucleic acid template. The aggregate absorption spectrum in the Soret region is resolved into two bands of Lorentzian line shape, while the DNA-bound monomer spectrum in this region is composed of two Gaussian bands. The Beer-Lambert law is obeyed by both porphyrin forms. The assembly is also characterized by an extremely large, bisignate induced circular dichroism (CD) profile and by enhanced resonance light scattering (RLS). Both the CD and RLS intensities depend linearly on aggregate concentration. The RLS result is consistent with a model for the aggregates as being either of a characteristic size or of a fixed distribution of sizes, independent of total porphyrin concentration or ionic strength. Above threshold values of concentration and ionic strength, the mass action expression for the equilibrium has a particularly simple form: K' = cac-1; where cac is defined as the "critical assembly concentration."offe dependence of the cac upon temperature and ionic strength (NaCl) has been investigated at a fixed DNA concentration. The value of the cac scales as the inverse square of the sodium chloride concentration and, from temperature dependence studies, the aggregation process is shown to be exothermic.     

The self-association of zinc-free bovine insulin. A single model based on interactions in the crystal that describes the association pattern in solution at pH 2, 7 and 10

Sedimentation equilibrium studies are used to establish that a new pattern for the self-association of zinc-free insulin in solution is applicable over a wide range of conditions of pH, ionic strength and temperature. In this pattern, which is based on information from the existing literature on the X-ray crystal structure of insulin, the insulin monomer is viewed as having two distinct faces both capable of self-interaction. Sedimentation equilibrium experiments were analysed using expressions formulated for this association pattern that describe the dependence of weight average molecular weight and monomer concentration on total protein concentration. It has thereby been possible to obtain values for the two association constants which govern the system for each set of conditions studied, due allowance having been made for composition dependent non-ideality effects. Furthermore, by relating the pH, temperature and ionic strength dependence of the association constants with properties of various amino acid residues on the surface of the insulin monomer, it has also been possible to assign tentatively each constant to a particular reaction domain.     

Modelling intravascular delivery from drug-eluting stents with biodurable coating: investigation of anisotropic vascular drug diffusivity and arterial drug distribution

In-stent restenosis occurs in coronary arteries after implantation of drug-eluting stents with non-uniform restenosis thickness distribution in the artery cross section. Knowledge of the spatio-temporal drug uptake in the arterial wall is useful for investigating restenosis growth but may often be very expensive/difficult to acquire experimentally. In this study, local delivery of a hydrophobic drug from a drug-eluting stent implanted in a coronary artery is mathematically modelled to investigate the drug release and spatio-temporal drug distribution in the arterial wall. The model integrates drug diffusion in the coating and drug diffusion with reversible binding in the arterial wall. The model is solved by the finite volume method for both high and low drug loadings relative to its solubility in the stent coating with varied isotropic–anisotropic vascular drug diffusivities. Drug release profiles in the coating are observed to depend not only on the coating drug diffusivity but also on the properties of the surrounding arterial wall. Time dependencies of the spatially averaged free- and bound-drug levels in the arterial wall on the coating and vascular drug diffusivities are discussed. Anisotropic vascular drug diffusivities result in slightly different average drug levels in the arterial wall but with very different spatial distributions. Higher circumferential vascular diffusivity results in more uniform drug loading in the upper layers and is potentially beneficial in reducing in-stent restenosis. An analytical expression is derived which can be used to determine regions in the arterial with higher free-drug concentration than bound-drug concentration.     

Rotational dynamics of transfer ribonucleic acid: effect of ionic strength and concentration

We have investigated the influence of ionic strength and nucleic acid concentration on the rotational Brownian motion of Escherichia coli tRNA1Val by studying the decay of the fluorescence polarization anisotropy (FPA) of intercalated ethidium on a nanosecond time scale. The rotational relaxation time tau R remains essentially constant as the ionic strength is varied from 2 to 100 mM at a tRNA concentration of 54 mg/mL. tau R also remains practically unchanged as the tRNA concentration is varied from 0.3 to 54 mg/mL at an ionic strength of 130 mM. Present hydrodynamic theories generally predict a more pronounced concentration dependence for rotational diffusion than we observe. This disagreement may result from a nonrandom distribution of the tRNA molecules in solution due to electrostatic interactions. By combining independent data from time-resolved nuclear Overhauser effect (NOE) cross-relaxation experiments and FPA experiments on the same tRNA, we are able to estimate the interproton spacing for the guanine N1-H and the uracil N3-H of the GU-50 base pair in E. coli tRNA1Val. This distance is 0.272 nm.     

How does vestibule surface charge affect ion conduction and toxin binding in a sodium channel?

We describe various models for the dielectric geometry and pore mouth charge distribution of a Na channel. The electric potential due to the vestibule charges is then computed on the basis of the nonlinear Possion-Boltzmann equation. The results are used to account for the effect of permeant ion concentration and ionic strength on channel conductance and on toxin association rate constants for Na channels. We find that a single negatively charged group near the entrance to the channel constriction is adequate to account for deviations from Michaelis-Menten conductance kinetics and for the concentration dependence of toxin-binding coefficients. We find further that only a limited range of vestibule geometries and pore mouth charge distributions are consistent with experiment.     

Ion dependence of the Bacillus subtilis RNase P reaction

The properties of the Bacillus subtilis RNase P are characterized with regard to the types and concentrations of monovalent and divalent ions required to potentiate precursor tRNA cleavage by the protein-RNA holoenzyme and the catalytic RNA alone. The ionic dependence of the RNase P RNA-catalyzed reaction in part seems due to a requirement for ion shielding between substrate and catalytic RNAs. The RNase P protein, which binds to RNA nonspecifically and tightly, likely serves, in part, as a cation screen. However, the character of the ion dependence of the RNA catalysis, the inhibition by high SO2-4 concentration, and potentiation by solvents suggest that RNA conformational transition may be involved in the reaction. It is proposed that the reason for catalysis by RNA in the RNase P reaction may be a requirement for fluidity in the structure of the catalyst, so that it can accommodate many tRNA substrates, which vary in their structural details.     

Kinetic characterization of the interaction between cytochrome oxidase and cytochrome c

The mechanism of electron transfer catalyzed by cytochrome oxidase was investigated by monitoring the reaction of cytochrome oxidase with cytochrome c under carefully controlled anaerobic conditions. The kinetics of the reaction were examined by varying conditions of ionic strength, inhibitor binding, and oxidation-reduction potential. An analogue of cytochrome c in which the iron atom was replaced with cobalt was used to probe the effect of redox potential on the reaction. Under conditions of low ionic strength, there is very rapid oxidation of cytochrome c and reduction of oxidase which occurs at a rate of 3 X 10(7) M-1 s-1. The number of electrons transferred exhibit a hyperbolic dependence on the concentration of cytochrome c reaching a maximum of 2 electrons transferred at the highest concentration of reduced cytochrome c employed. The total number of electrons transferred was always observed to be distributed equally between cytochrome a and a second acceptor which appears to be the associated copper center; electron transfer to cytochrome a3 did not occur in the absence of oxygen. Substitution of cytochrome c by the cobalt analogue (which represents a decrease in oxidation-reduction potential of about 400 mV) yielded identical results indicating that the origin of the lack of reactivity of cytochrome a3 is of a kinetic nature. The effect of increasing the ionic strength on the reaction was 2-fold: a marked decrease in reaction rate and the appearance of biphasic kinetics with the amplitude of the very fast absorbance changes at 605 nm decreasing from 80% to 40% of the total anticipated from static absorbance measurements. Each of the two phases accounted for a maximum of 1 electron at the highest ionic strength employed. These results are simulated in terms of a sample kinetic reaction scheme involving a two-step electron transfer at one binding site.     

Electrostatic effects on the kinetics of bound enzymes.

1. The effect of the interaction between the charged matrix and substrate on the kinetic behaviour of bound enzymes was investigated theoretically. 2. Simple expression is derived for the apparent Km. 3. The apparent Km can only be used for the characterization of the electrostatic effect of the ionic strength does not vary with the substrate concentration. 4. The deviations from Michaelis-Menton kinetics are graphically illustrated for cases when the ionic strength varies with the substrate concentration. 5. The inhibition of the bound enzyme by a charged inhibitor at constant ionic strength is characterized by an apparent Ki. 6. When both the inhibitor concentration and the ionic strength change there is no apparent Ki, and the inhibition profile is graphically illustrated for this case. 7. Under certain conditions the electrostatic effects manifest thenselves in a sigmoidal dependence of the enzyme activity on the concentration of the substrate or inhibitor.     

A steady-state study on the formation of Compounds II and III of myeloperoxidase

The reaction between native myeloperoxidase and hydrogen peroxide, yielding Compound II, was investigated using the stopped-flow technique. The pH dependence of the apparent second-order rate constant showed the existence of a protonatable group on the enzyme with a pKa of 4.9. This group is ascribed to the distal histidine imidazole, which must be deprotonated to enable the reaction of Compound I with hydrogen peroxidase to take place. The rate constant for the formation of Compound II by hydrogen peroxide was 3.5.10(4) M-1.s-1. During the reaction of myeloperoxidase with H2O2, rapid reduction of added cytochrome c was observed. This reduction was inhibitable by superoxide dismutase, and this demonstrates that superoxide anion radicals are generated. When potassium ferrocyanide was used as an electron donor to generate Compound II from Compound I, the pH dependence of the apparent second-order rate constant indicated involvement of a group with a pKa of 4.5. However, with ferrocyanide as an electron donor, protonation of the group was necessary to enable the reaction to take place. The rate constant for the generation of Compound II by ferrocyanide was 1.6.10(7) M-1.s-1. We also investigated the reaction of Compound II with hydrogen peroxide, yielding Compound III. Formation of Compound III (k = 50 M-1.s-1) proceeded via two different pathways, one of which was inhibitable by tetranitromethane. We further investigated the stability of Compound II and Compound III as a function of pH, ionic strength and enzyme concentration. The half-life values of both Compound II and Compound III were independent of the enzyme concentration and ionic strength. The half-life value of Compound III was pH-dependent, showing a decreasing stability with increasing pH, whereas the stability of Compound II was independent of pH over the range 3-11.     

Concentration-dependent isomerization of bacteriophage T2L

We have studied the concentration and ionic strength dependence of the fiber-reorientation transition in T2L bacteriophage, using sedimentation velocity and quasielastic light scattering diffusion measurements. High-salt and low-phage concentration favor the form with fibers extended. The equilibrium between the two forms is dependent on the phage Concentration. This behavior is unexpected for a reaction that is apparently unimolecular with respect to phage, and attests to the strong increase in excluded volume and other interactions accompanying fiber extension. The fiber-extended form has anomalously high diffusion coefficient and low molecular weight according to the Svedberg equation.